Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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3
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1answer
536 views

Two-Point boundary value problem

To solve ${d^2y \over dx^2} =f(x)$, $0<x<1$ with $y(0)=\alpha, y(1) = \beta$. We can get a finite difference approximation by taking $$\frac{y_{j+1}-2y_j+y_{j-1}}{h^2} =f_j \\\Rightarrow \frac{1}...
3
votes
2answers
145 views

Solve the pde $u_t(x,t)=u_{xx}(x,t)-bu(x,t)+q(t)$ for $u(x,t)$

I have the example pde $u_t(x,t)=u_{xx}(x,t)-b(t)u(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. The subscripts denote derivatives. I also have ...
3
votes
2answers
109 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ c)...
3
votes
1answer
3k views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
2
votes
2answers
77 views

second order DE using reduction of order

Any Hints / details on how to find a second solution for $$x^2y'' + xy' -4y=0?$$ $$y_1 = x^2 y_2$$ I need to use reduction of order thanks
2
votes
0answers
71 views

Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff y=\left(x+C\right)^{...
2
votes
1answer
1k views

Solving a partial differential equation using method of characteristics

I keep getting stuck and have a hard time understanding my professor, so I'm hoping to get some help here. The question is: Solve the partial diff/eq: ${\partial u}\over{\partial t}$ + $c {{\partial ...
2
votes
1answer
194 views

Finding a value a for topologically conjugacy between two flows

Let A be a hyperbolic matrix such that all solutions of $\overrightarrow x' = A \overrightarrow x $ tend to the origin at t goes to infinity, and suppose B = $\begin{bmatrix}a-3 & 5 \\ -2 & a+...
2
votes
0answers
161 views

Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
2
votes
1answer
67 views

Differential Equation - $y'=|y|+1, y(0)=0$

The equation is $y'=|y|+1, y(0)=0$. Suppose $y$ is a solution on an interval $I$. Let $x\in I$. If $y(x)\ge 0$ then $$y'(x)=|y(x)|+1\iff y'(x)=y(x)+1\iff \frac{y'(x)}{y(x)+1}=1\\ \iff \ln (y(x)+1)=x+...
2
votes
2answers
642 views

Considering the linear system $Y'=AY$

What would be an equation that I can use when I compute the eigenpairs for the coefficient matrix $A.$
2
votes
3answers
908 views

Express differential equations as system of first order equations

Express the differential equation $$y'''-6y''-y'+6y=0$$ as a system of first order equations i.e. a matrix equation of the form $$A(\vec x)'=0$$ where $$\vec x\text{ is the vector }\left[ \begin{...
2
votes
2answers
81 views

Procedure for 3 by 3 Non homogenous Linear systems (Differential Equations)

Here is the problem I have. $$x^{'}(t)=Ax(t)+f(t)$$ where $A=\begin{pmatrix} 5&-3&-2\\8&-5&-4\\-4&3&3 \end{pmatrix} f(t)=\begin{pmatrix} -\sin (t)\\ 0 \\ 2 \end{pmatrix}$ I am ...
2
votes
1answer
72 views

Specific system of differential equations

I have the following system of equations: \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray} I want to prove that if a solutions starts (at time $t = ...
2
votes
1answer
300 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: $\frac{d\mathbf{x}}{...
2
votes
1answer
90 views

Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
2
votes
2answers
127 views

Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
2
votes
2answers
1k views

How can I prove that the DE $y'=y^\alpha$ has infinitely many solutions?

I need to show that the DE $y'=y^{\alpha}$, where $\alpha$ is a constant with $0<\alpha<1$, has infinitely many solutions passing through the point $(0,0)$. Also I need four of such solutions. ...
2
votes
1answer
143 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
2
votes
1answer
80 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on $\mathbb{R}^3$...
2
votes
3answers
156 views

Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods

I have to calculate approximations of the solution with the method $$ y^{n+1}=y^n+h \cdot [\rho \cdot f(t^n,y^n)+(1-\rho) \cdot f(t^{n+1},y^{n+1})] ,\quad n=0,\ldots,N-1 \\ y^0=y_0 $$ for various ...
2
votes
2answers
684 views

Generalized “Worm on the rubber band ” problem

I found this « Worm on the rubber band » problem in Concrete Mathematics book. A slow worm $W$ starts at one end of a meter-long rubber band and crawls one centimetre per minute toward the other end. ...
1
vote
2answers
130 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: $d_{k+1}=u(t_{k+1})-\underset{u_{k+1}}...
1
vote
2answers
92 views

Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
1
vote
1answer
336 views

Multistep ODE Solvers

Write both a fourth order Adams Bashforth and Adams Moulton procedure to solve $$x'(t) = x(t)-y(t)-\exp(t);$$ $$y'(t) = x(t)+y(t)+2\exp(t)$$ with initial values $x(0) = -1, y(0) =- 1$ on the ...
1
vote
2answers
32 views

Differentiaion Calculus: Trig Inverse Function.

Well, yesterday at a Mathematics exam, i had to find $\frac{dy}{dx}$ of a cotangent inverse function $$ y=\text{arccot}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-sin x} }\...
1
vote
2answers
190 views

$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$

Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\begin{cases}1 & \text{ if } 0\le x\le 1\\0 & \text{ if } x>1\end{cases}$$Then, $...
1
vote
1answer
73 views

Modelling population with $\frac{dP}{dt}=P(\beta - \delta P)$

The population $P(t)$ of a biological species can be modelled by $$ \frac{dP}{dt}=P(\beta - \delta P) $$ subject to $P(0)=P_0$ where $\beta$ is the birth rate and $\delta$ is the death rate. ...
1
vote
2answers
403 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } |y-e|\leq\frac{...
1
vote
1answer
93 views

Proof of existence and uniqueness of the exponential function using ODEs

In our lecture notes for our complex analysis class, we were given the following theorem: Theorem: There exists a unique complex function $f$ such that $f(z)$ is a single valued function $...
1
vote
2answers
724 views

Using Octave to solve systems of two non-linear ODEs

How to solve following system of ordinary differential equations using Octave? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ Update: initial conditions: $x(t=...
1
vote
2answers
50 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} \...
1
vote
3answers
370 views

Tricky Separable Differential Equation

Please guide me: $y' + ay +b = 0$ ($a$ not zero) is supposed to be separable and has solution $y = ce^{-ax} - \frac ba$ Here is my start to this problem: $\frac{dy}{dx} + ay = -b$ is as far as I ...
1
vote
0answers
14 views

IVP Using Numerical Methods

Suppose that $y(t)$ is the exact solution of the ivp $$y'(t)=f(t,y(t)), y(0)=y_0$$ and $u(t)$ is any approximation to $y(t)$ with $u(0)=y(0)$. Define the error $e(t)=y(t)-u(t)$. How can I show that ...
1
vote
3answers
106 views

Solve $y'-\int_0^xy(t)dt=2$

I have not idea how to approach this differential equation. $$y'-\int_0^xy(t)dt=2$$. Basically, I did, $$F''(t)-F(x)+F(0)=2 \;\;\;\;\;\;\; F'=y$$ I am stuck. Thank You.
1
vote
1answer
85 views

Solutions and attraction regions of following odes?

Assume a mapping $X: \mathbb{R} \to \mathbb{R}^d$. We know that the solution to ode $$ d X_t = (\mu - X_t) dt $$ is $X_t = (X_0-\mu) e^{- t} + \mu$, which indicates that $X_t$ converges to $\mu$ as $...
1
vote
1answer
46 views

general conditions for reverse poincare inequality

I'd like to know when the reverse Poincare inequality is true: Given a bounded domain $\Omega$, and $f \in L^2(\Omega)$, under what conditions on $f$ (neglecting the trivial constant case) and/or $\...
1
vote
2answers
5k views

Difference between improper node and proper node for phase portrait

Can someone offer an explanation for the difference between these two? I see pictures of what seem to be examples of both, but it's hard for me to discern what a new portrait would be. Any help? ...
1
vote
3answers
456 views

General Solution for a given system of equations

Find the general solution of this system of equations: $$x' = \pmatrix{-1&0&0\\1&0&-1\\1&1&0}x$$ I got the eigenvalues to be: $\lambda = -1,\pm i$ The eigenvectors ...
1
vote
3answers
573 views

Is $f'(x)-3f(x) = 0$ subspace of differentiable functions $f\colon (0,1)\to \mathbb{R}$

$V$ is space of differentiable functions $f(0,1) \to \mathbb{R}$ and $W$ is a subset of $f$ that meets $f'(x) - 3f(x) = 0$ for all $x\in (0,1).$ Is subset $W$ a subspace of $V$? I know that I have ...
1
vote
1answer
47 views

H0w are second order nonlinear ordinary differential equations solved?

I conceived the following second order nonlinear ordinary differential equation: $$\frac{d^2y(x)}{dx^2}=\frac{k}{(y(x))^2}$$ I can tell it's nonlinear because of the $\frac{k}{(y(x))^2}$ term and ...
0
votes
1answer
81 views

Initial Value Problem

Please help solving the following initial value problem: $$y''-3y'+2y \; = \; 3e^{-x}-10 \cos {3x}; \;\;\; y(0)= 1, \;\;\; y'(0)=2 $$ I have been working at it and have been hitting a road block
0
votes
2answers
69 views

trouble with non-homogeneous ODE system… which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ...
0
votes
1answer
817 views

Finding the interval where a solution is certain to exist for the equation $y' + (\tan t)y = \sin t$

Given the following problem: Determine (without solving the problem) an interval in which the solution is certain to exist for the initial value problem $y' + (\tan t)y = \sin t, \space y(2\pi) = ...
0
votes
3answers
336 views

Differential Equations Skydiver Problem

I've seen many variants of this problem online, but not quite the same as this, so I don't believe this is a duplicate. The famous differential equation problem models a skydiver jumping out of a ...
0
votes
1answer
137 views

How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.

Let $f:[0,1]\to\Bbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0,1)$ and $f(0)=f(1)=0$. Then the equation $f(x)=f'(x)$ admits how many solutions? The only solution that ...
0
votes
2answers
34 views

Find the first order system of linear equations

Regard the diff equation: $mϕ′′+aϕ′+(mg/L)ϕ=0$ $ϕ(0)=0.1$ $ϕ′(0)=0$ where $m=0.1,L=1,a=2,$ 1) Rewrite the second order diff equation as a system of first order linear equations. 2) What is the ...
0
votes
6answers
155 views

Solution to a second order ordinary differential equation

Let $\beta >0$, $\gamma > 0$, $\omega > 0 $ and $\xi >\xi_0 $. The question is to show that the solution to the following inhomogenous ordinary differential equation: \begin{equation} \...
0
votes
1answer
128 views

Find the velocity of a flow

The question is: Find the velocity of the flow described by the velocity potential given in the polar coordinates $φ$$(r, θ)$ = $θ$, where $x = r cos θ$ and $y = r sin θ$, $r > 0, 0 ≤ θ < 2π$ ...
0
votes
1answer
53 views

Bring the linear part of a ODE system into a diagonal form

Consider the system $$ \dot{x}=y,\quad\dot{y}=z,\quad\dot{z}=-x-y-z+x^2+az^3. $$ Check that the zero equilibrium has a pair of pure imaginary eigenvalues. Make a linear coordinate ...